Maximum-a-posteriori (MAP) estimation. Suppose there are a set of finite classes \(\mathcal {C}\) . A class label \(Y\in \mathcal {C}\)  is generated with probability \(\pi _Y\) , where \(\sum _{Y\in \mathcal {C}} \pi _Y = 1\) . Given \(Y\) , the corresponding feature \(X\)  in the space \(\mathcal {X}\)  is generated from the distribution \(X\sim \mathbb {P}_{Y}\) . A classifier is a decision \(f:\mathcal {X}\rightarrow \mathcal {C}\)  and the Bayesian mis-classification error can be characterized as \(\xi (f)= \sum _{Y\in \mathcal {C}} \pi _Y \int 1_{f(X)\ne Y} \mathbb {P}_{Y}(X)\) , where and later \(1_{S}\)  indicates 1 if \(S\)  is true and 0 otherwise. The MAP estimation of \(Y\)  given \(X\)  is the classifier \(f^*(X) \triangleq \arg \max _{Y\in \mathcal {C}} \pi _Y \mathbb {P}_{Y}(X)\)  that can minimize \(\xi (f)\)  [1]}. Later, we denote the minimal Bayesian mis-classification error \(\xi (f)\)  as \(\xi ^* = \xi (f^*)\) .
